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Have you ever been told that division by zero is undefined as there is no number for infinity?

Well zero is also an imaginary concept. If I were to say that I have zero red, white, and blue dogs in my house that would be a lie. The fact that I imagined such a creature gives it a non-zero quality.

The image or concept of a red white and blue dog, is not the same thing as a real red white and blue dog. (See Rene Magritte's La trahison des images.) Confusing the two would be a category error. Being able to imagine the hypothetical or the contrafactual is a unique trait of humanity over other animsls, one that confered upon us a significant survival value, because it allowed us to bring many useful tools into our world that did not exist previously. But being able to distinguish between the imaginary and the real is also an essential survival trait and a sign of a healthy mind. Confusing the two is a sign of derangement, or deliberate subterfuge, or both.

Corrollaries:

"The word is not the thing.""The map is not the territory."

There are still zero red white and blue dogs in your house. Zero is a perfectly workable and useful number, within limits. Thankfully, our culture has been out from under the oppression of the Pythagorean cult for many centuries.

The image or concept of a red white and blue dog, is not the same thing as a real red white and blue dog. (See Rene Magritte's La trahison des images.) Confusing the two would be a category error. Being able to imagine the hypothetical or the contrafactual is a unique trait of humanity over other animsls, one that confered upon us a significant survival value, because it allowed us to bring many useful tools into our world that did not exist previously. But being able to distinguish between the imaginary and the real is also an essential survival trait and a sign of a healthy mind. Confusing the two is a sign of derangement, or deliberate subterfuge, or both.

Corrollaries:

"The word is not the thing.""The map is not the territory."

There are still zero red white and blue dogs in your house. Zero is a perfectly workable and useful number, within limits. Thankfully, our culture has been out from under the oppression of the Pythagorean cult for many centuries.

I see that you disagree with my concept. I believe that to imagine something makes it a tiny bit real. You obviously don't. I don't think it possible to imagine something unless it is in some way real. It does not matter that a battleship cannot even fit inside my house. If enough people imagine a red white blue dog someone might buy me a pillow that looks like a dog or a plastic toy.

I see that you disagree with my concept. I believe that to imagine something makes it a tiny bit real. You obviously don't. I don't think it possible to imagine something unless it is in some way real. It does not matter that a battleship cannot even fit inside my house. If enough people imagine a red white blue dog someone might buy me a pillow that looks like a dog or a plastic toy.

Disagreement is a good thing as it can lead to a higher truth.

There is a concept I learned in calculus. Limit approaching zero is zero for practical purposes, but there is a huge difference between the limit and actual zero.

There is a path of events between actual zero and one of anything as I see things. Suppose I didn't have even the thought of a dog in my house then I get thoughts of a dog in my housethen I get the intention of getting a dogthen I learn my friend's dog had puppiesthen I get a puppythen the puppy becomes an adult dog.

Philosophically speaking, 0 is as real as any number. What sets 0 and infinity (which isn't a number) apart is that they are not tangible; but then neither is any negative number. The idea of a tangible number semifield limits the numbers to the positive reals. I wanted to excise all the non-tangible 'craziness', but that comes at the price of handicapping the tool of mathematics. In science, even the entire real field is not enough; so much problem-solving is hobbled without the complex numbers. There's a cautionary tale here.

Many years ago I stumbled upon a website called MetaResearch, by the late astrophysicist Tom Van Flandern. One article caught my interest, named 'Physics Has Its Principles', where he lamented the growing craziness of physics caused by overreliance on mathematics, where, he said, 'anything goes' and an immediate connection to reality is optional. Van Flandern proposed a modified Steady State theory of the universe where those consequences of General Relativity and Quantum Mechanics that go against common sense are corrected, such as the idea that something can be in two places at the same time, and the Big Bang itself.

Now, I'm by no means qualified to judge ideas on physics (though I ought to note that none of Van Flandern's ideas on 'deep reality' have been accepted by the mainstream), but upon sinking those ideas inside, I felt a mixture of agreement and disagreement: on the one hand, I'm glad an actual scientist has turned his attention to the problem with current physics, but on the other, I could only say to myself, 'That boat (of science matching common sense) sailed away long ago'. It sailed away when the ancient Greeks realised we're standing on a ball, with the possibility of people 'hanging by their feet' on the other side.

Yes, physics makes my head spin with its plethora of counterintuitive findings. But biology has plenty of its own such findings, yet I enjoy studying biology in a way I can't say about physics. Why the difference? In all probability, it's because biology has a grand unifying principle (the neo-Darwinian evolutionary-genetic synthesis theory), while physics is still struggling with the discrepancy between its small-scale (quantum) and large-scale (relativity) planks (Plancks? ). I suspect biology must have been just as confusing before Darwin, and I have hope physics will be plain sailing once it gets its own grand unifying theory. But throwing out the baby of common sense with the bathwater of confusion isn't the way to go, which is why Van Flandern hasn't found any more favour than William Dembski (of intelligent design notoriety).

The moral here is that tangible reality and common sense ideas are not equal to utility or even the whole of reality. Negative numbers are intangible, and complex numbers (based on i, the square root of −1) are doubly so; but the field of complex numbers is the one under which all polynomials have a solution, which is why scientific mathematics would grind to a screeching halt without them, and with it, technological progress, which is something very tangible to all of us.

There are still zero red white and blue dogs in your house. Zero is a perfectly workable and useful number, within limits. Thankfully, our culture has been out from under the oppression of the Pythagorean cult for many centuries.

Zero is pretty definate. That is, it's a natural number describing the absence. One might recall that the greeks held the numbers began with two.

Infinity is a different story.

I use the model of the small infinity, where the object of the game is not to make infinity as large as possible, or as inclusive as possible, but as large and as inclusive as needed. This is largely the model of the dominate part of equations, or eventually, that bits that are otherwise diverse (as euclidean, möbius and hyperbolic geometry), are the same thing at different scales.

Mind you, the model of the small infinity makes infinity a subset of a larger set of indefinites, some of which are zero-like. Of course, this is an outcome of log(log(x)) etc, and log(x), where the hyperbolic horizon is infinite in appearence, while perfectly reachable (eg 80 digits vs 80). The whole point is that the cascade infinities tells us that we can reach very large numbers with a small number of free variables. Yet the cascade infinities are class-2, the smallest of the infinities.

Zero is pretty definate. That is, it's a natural number describing the absence. One might recall that the greeks held the numbers began with two.

No doubt, if the Germanic peoples held that, it would be touted here as some great insight.

QUOTE (wendy.krieger @ Mar 23 2017, 09:58 AM)

Infinity is a different story.

The concept of infinity is pretty definite. It simply describes something that is unbounded.

QUOTE (wendy.krieger @ Mar 23 2017, 09:58 AM)

I use the model of the small infinity, where the object of the game is not to make infinity as large as possible, or as inclusive as possible, but as large and as inclusive as needed. This is largely the model of the dominate part of equations, or eventually, that bits that are otherwise diverse (as euclidean, möbius and hyperbolic geometry), are the same thing at different scales.

Mind you, the model of the small infinity makes infinity a subset of a larger set of indefinites, some of which are zero-like. Of course, this is an outcome of log(log(x)) etc, and log(x), where the hyperbolic horizon is infinite in appearence, while perfectly reachable (eg 80 digits vs 80). The whole point is that the cascade infinities tells us that we can reach very large numbers with a small number of free variables. Yet the cascade infinities are class-2, the smallest of the infinities.

Ignoring the usual spate of undefined terminology, there is nothing of that sort of choosing the size of infinity. From the analytical sense (which is presumably what is intended here, not the set-theoretical sense with multiple levels of infinity; those are not symbolised as ∞ anyway), every complex number corresponds to a set of sequences which converge to that complex number as their limit, and the improper complex number ∞ corresponds to those sequences which are unbounded and diverge towards infinity: given any positive number ρ, you will be able to find a natural number n such that the moduli of all terms of the sequence after the nth are greater than ρ. There is no choosing of the size of infinity here: ∞ is not a number you can have in the sequences, so there can't be a sequence converging to something "more than infinity", whatever that might mean. Likewise, every other improper complex number has a magnitude lower than that of ∞. So you can't increase the size of ∞, nor decrease it.

Zero is not a counting number, but a natural number. The Greeks only held view that numbers were plural, and that one also was not a number. But mathematically speaking, it is a natural number.

Infinity is best dealt with by avoiding it, or failing that, by supposing something like gradient without coordinate. In any case, Euclid's geometry makes no use of infinity, it is in essence fE, where f, o, and s are by rejecting one statement of "in a complete (c ) orientable <s> space, lines cross once <o>" oS is orientable complete spherical, while sS is single-crossing complete spherical (elliptic). oE is the Mobius geometry.

Of course, if you suppose infinity is finite, the true nature of the euclidean horizon is revieled. It's a complete euclidean geometry oE. It's easy to prove this, by noting that AB = BA, applies to infinite lines as well, so the locus of the centres of isocurves passing through A, is an isocurve of the same nature centred on A.

From a physics point of view, you can choose the size of infinity. The smallest teelic infinity is 71. It's not known what the second one is. 881 sounds about right. This is because the number of paths is less than the number of outcomes, and so '2+2', '3+1' and '4' are all paths to the same number. Numbers that have no single digit have a path of the digits. So x17 = z15 are alternate paths to the seventh prime.

Since infinity is observable, one can understand the shortcomings of the cantor models. The diagonal proof falls apart, simply by definition, you can not list all of the numbers, since the count of the digits used can only ever cover the last digit. It fails in any case for bases less than two.

The fact it raises so many paradoxes, is simply because the assumptions are wrong. The numbers never break down, in that if you suppose A(0) = a, A(n+1)=b^A(n), can always be reduced modulo x, for any given x, it converges. The order of convergence is something like a base-tail (P-adic) in base b to n-c places (c a small value). It then happens that thus that A(n)+1 is always distinguishable from A(n).

What happens is that the operators become fuzzy. It's not perfectly strict, but we see only certain properties of a number appearing, such as its mantissa relative to b, or its modulus, or something like this. The example of using a real point to represent infinity, and projection through lattices, is a reduction relative to mantissa 2 or 3.

It is therefore, seemingly possible to bounce things off the same infinity, if parts of its construction is known. In one feature, this is the result of Euclidean geometry, which relies on dividing infinity by infinity, and 0/0, but with known constructions.

If there were any doubt left that Wendy is a crank, it's all disappeared with her rejection of Cantor's diagonal argument (which is very common among cranks; it is usually the first bit of nonsense that issues from them the moment they stop being humble enough to realise that they might not be right), and the nonsensical final statement. Surely we would have realised by now if nonsense like ∞/∞ and 0/0 was coming up in Euclidean geometry. Actually it doesn't.

I would give a full rebuttal, but why bother: it's not going to be effective for her, and the argument is so famous that those who are interested in actual mathematics (instead of her pseudomathematics) can go look it up for themselves and be appropriately awed.

If there were any doubt left that Wendy is a crank, it's all disappeared with her rejection of Cantor's diagonal argument (which is very common among cranks; it is usually the first bit of nonsense that issues from them the moment they stop being humble enough to realise that they might not be right),

Not to mention her assertions that Einstein's theories of special and general relativity can be dismissed with a bit of 11zst-biquennium (19dth-century) vintage armchair algebra...

If there were any doubt left that Wendy is a crank, it's all disappeared with her rejection of Cantor's diagonal argument (which is very common among cranks; it is usually the first bit of nonsense that issues from them the moment they stop being humble enough to realise that they might not be right),

Not to mention her assertions that Einstein's theories of special and general relativity can be dismissed with a bit of 11zst-biquennium (19dth-century) vintage armchair algebra...

What DoubleSharp is not seeing, is that the description i give is a fairly typical presentation of physics, where the 'depth' of the number is roughly as zipf's law.

The idea is that for small numbers you can 'know' their variations. For a number like 100,000, or the like, you can visit all the instances to it. For a number of thirty digits, you can know random examples, without being able to visit. And so on.

The idea that Cantor proposes is that you can 'count' to any namable number, and that any number that has a name is countable. But the argument that the list that supposedly contains all numbers, can not do so, since it takes more numbers to express it, so the proposition they were trying to fill is false even before they commit the diagonal.

A small infinity of 10^80, would express in just eighty digits, and so the diagonal runs out well before the count does. And the further one extends the digits, one digit more gives ten times the examples.

From this, we are supposed to believe that at some point a = 10^a for this comparison to work. The proof is that a < 10^a, but the proof fails miserably where there is an existance of exclusion rules, such as in base phi.

More significantly, the teelic infinities tell us that the number of paths is necessarily greater than the destinations, and that even the diagonal might constitute a path to a known number, in a different expression. So the necessary implication is something like "4 is not on the list because '2+2' is different to the diagonal, but '3+1' is, and so 4 is on the list, but not on the path".

The construction of the diagonal is itself a cascade form in a single direction.

It is interesting that Double-Sharp mentions 'germanic nationalism' in his comment on the first post. Much of this 'crank' stuff he goes on about is rather much the same as any other cultural superiorism, that is, the 'not invented here'.

I have the foresight, to realise what i have done, is a different culture to the standard method, and understand that this is due to different perceptions, rather than right vs wrong.

The innumeracy inherent in Wendy's statement against Cantor's proof, that it doesn't work for numbers expressed in a non-integer base, is that there is no difference between the reals expressed in one base and the reals expressed in another base. They are the same set of numbers, just written differently. So if you've proved that the reals written in decimal are uncountable, then so are the reals written in dozenal, the reals written in octal, the reals written in twelfty, the reals written in base pi, the reals written in base phi, or the reals written in base gamma (the Euler-Mascheroni constant), or whatever.

It is very funny how she seems to think that changing the base means that a might not be less than 10a (of course, for appropriately large a), which is another case of pure innumeracy, and valuing the written form over what it might mean. (Then again, perhaps we should have seen this silliness coming, since she wants to call "10" "ten" regardless of what it actually means.)

The supposed "problem" that Wendy detects is simply a failure to understand proof by contradiction. Cantor is simply saying:

1. Assume that there is a one-to-one correspondence between members of N and members of R.2. Note that there must be members of R that do not correspond to any member of N.3. This contradicts the assumption that the correspondence was one-to-one. So there can be no one-to-one correspondence between N and R and R is uncountable.

I don't disdain your work because I didn't invent it. I'm not somehow jealous of it; if I had come up with it, I'd be ashamed of it. I, unlike you, have the humility to understand that the standard method is standard for a reason, and in a subject like mathematics, it is not a case of different perceptions, but of you just being clueless.

In fact, I'm quite done with you. For a while I laboured under the misapprehension that you actually had some nuggets of sensibility going on under all the obfuscation. Now, when the obfuscation is done away with, we have an epitome of cranky pseudomathematics of zero value.

Now I can ignore your ramblings with a clear conscience that I'm not missing anything that might be useful.

What DoubleSharp is not seeing, is that the description i give is a fairly typical presentation of physics, where the 'depth' of the number is roughly as zipf's law.

The idea is that for small numbers you can 'know' their variations. For a number like 100,000, or the like, you can visit all the instances to it. For a number of thirty digits, you can know random examples, without being able to visit. And so on.

Once again, Wendy conflates concepts from linguistics (or more broadly, statistical behavior of data sets) with concepts from pure mathematics and physics. I'm tempted to characterize this habit of hers as a kind of deepity, which is an attempt to sound deeply profound and wise, by making statements that at a literal level are true but trivial, but at a figurative level are false, but would be earth-shattering if they were true. Except that Wendy's utterances tend not to be true even at a trivial literal level.

Zipf's law was originally an empirical statistical observation about the frequency of usage of words:

QUOTE (Wikipedia)

Zipf's law states that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table.

This observation has absolutely nothing to do with the mathematical concept of infinity, nor does it say anything to either support or refute Cantor's Theorem. She seems to believe that name-dropping something like "Zipf's law", without elaborating on it, might awe and impress the less discerning folks in the audience.

QUOTE (wendy.krieger @ Mar 25 2017, 09:07 AM)

The idea that Cantor proposes is that you can 'count' to any namable number, and that any number that has a name is countable. But the argument that the list that supposedly contains all numbers, can not do so, since it takes more numbers to express it, so the proposition they were trying to fill is false even before they commit the diagonal.

A small infinity of 10^80, would express in just eighty digits, and so the diagonal runs out well before the count does. And the further one extends the digits, one digit more gives ten times the examples.

Ten-to-the-eightieth-power is a finite number. Figuratively calling it a "small infinity" might make for a fun bit of poetry, but just because some bard might utter such a metaphor does not make ten-to-the-eightieth-power an "infinity" in any actual mathematical sense.

Even if it is practically impossible for one human to recite the first ten-to-the-eightieth-power whole numbers (or even the entire human race, even if they had the entire history of the universe to do so), that set of whole numbers is still a finite set, and is still countable in principle. What Wendy doesn't get is that you cannot use the same reasoning that applies to finite sets to reason about infinite sets. Yet it is eminently possible for human beings to reason about infinite sets, and distinguish between countably infinite versus uncountably infinite sets. But because Wendy doesn't get Cantor's Theorem, she assumes that means that it can't possiby be true. That's a logical fallacy known as an argument from personal incredulity, or more broadly an argument from ignorance.

QUOTE (wendy.krieger @ Mar 25 2017, 09:07 AM)

It is interesting that Double-Sharp mentions 'germanic nationalism' in his comment on the first post. Much of this 'crank' stuff he goes on about is rather much the same as any other cultural superiorism, that is, the 'not invented here'.

Once again, Wendy is projecting her own faults on others. She's demonstrated quite frequently her pro-Germanic cultural bias, and her inability to accept anybody else's ideas unless she can somehow claim to have thought of them first. Several of us here have observed this behavior and criticized it. But instead of addressing the criticism, and amending her behavior, she seeks to deflect it back onto her critics.

QUOTE (wendy.krieger @ Mar 25 2017, 09:07 AM)

I have the foresight, to realise what i have done, is a different culture to the standard method, and understand that this is due to different perceptions, rather than right vs wrong.

Ah, and here we get down to it: the inevitable self-aggrandizement. What Wendy doesn't understand is that a population of "one" is not a "culture". While it is bigoted to make disparaging generalizations about an entire ethnic group (something Wendy apparently thinks she can engage in with impunity), it is completely legitimate to criticize the flawed thinking of one individual person.

The whole point of Cantor's argument, is that if you have a 'complete' set of numbers derived by digits, then the variation of the diagonal would imply that it's not complete.

The fault is that a complete set of items is by itself not possible, and even if you suppose Cantor's proposition to be true, the number of digits used to express it is N². It is like saying 11 > 10, because we wrote 100 digits, and we could write a 10-digit number by differing the digits of the diagonal, and so there must be an 11th 10-digit number. The fault with the proposition is that it supposes that N² converges on N.

DoubleSharp's characterisation of what is the issue is simply false. The fault in the argument is that you can construct a complete constructable list out of a cascade series. In one hand, you are saying, 'there are 10 numbers' representing 'all the countable numbers'. But you use 100 digits to write these out to prove that there must be an eleventh one. This is what the physicists are objecting to.

Zipf's law applies to a large range of things. It applies to many things, like the size of cities and companies, for example. It is something related more to how humans choose things, and organise things for example. Its relevance here is described in the paragraph that followed its mention.

The insights that thinking about mathematics in physical terms, leads to the a kind of geometry that encloses the euclidean, hyperbolic, and spherical geometries, to the extent that it is wholy consistent with the nature of the universe (which is not known to be any of the three, because the error size of the point U crosses the surface of the conformal sphere). Straight lines are represented by the intersection of the sphere and a plane passing through U.

Since a good deal of the geometry is about deciding whether the coordinates of a figure lie in this set or that, and that whether the isomorphic forms of the set have the appropriate curvatures, the study of the class-rule, and the various kinds of infinity is used to cut out the crap.

Mathematicians suppose that there is no gap between the reality and the actuality. We spent a good deal of time discussing the particular issue, to the extent that a mathematical object does not exist until it's created as a separate thing.

The fact one sees things differently, is not itself different to the sorts of rationalisation one reads in foreign and older mathematics, that when one looks at say, indian mathematics or chinese mathemstics from european eyses, or whatever, it's more like surveying the countryside from the window of a passing train: you see things you recognise, but the order and insight is missing.

The people who are educated in technical colleges, rather than universities, tend to have a deeper understanding of the theory of the matter at hand. This is an observation made today on Quora, by an american about german engineers. Those who study in the universities, study a wider range of studies, but to a lesser depth than those who study at the technical colleges, who have a deeper understanding of the matter at hand.

In the question of 'german nationalism', it is worth noting that it was D.S. that first suggested the notion that 'had the germanics discovered zero, it would be a deep insight'. The art that i study, suggests some five different kinds of zeros, each in three representations. Likewise, i realised long ago, that i was passing over a large amount of territory that was uncharted, but many of the equations have been mentioned in passing. In essence, it is a track not often explored.

One might note that my paper lists some six or seven new uniform hyperbolic tilings in various dimensions.

When John Conway was writing his recent symmetry book, John and I discussed his new notation for describing hyperbolic tilings in terms of an addendum to the orbifold notation that he added to Thurston's discovery. We used quite different approaches, but came to much the same result. His model was more inclusive, but as i pointed out to him, it was too inclusive, and there was more charf than grain.

The method i used was to see if the methods of Wythoff and Coxeter could be applied to the symbol, but it proved to be a considerable nightmare, because the uniform polyhedra implements both miricales and wanders. John thought we could eliminate wanders by doubling up on miricales, but i found a counter-example.

And then there is Marek Ctrnack's discovery of (5,5,5,3). People tried to prove lots of things about it. Prof Johnson tried to link it to a horocyclic cascade, the order of which i calculated as Ø². We calculated the vertex-ratios and face-counts. It was suspected to be irrational, until i showed its rationality.

Of course, we can mix it with the best, so it's hardly crankery, but a whole new art of seeing things.

The inability to detect sarcasm is a mental condition, such as Aspegaries Condition.

The failure to detect that the sarcasm has misfired is something like insenstivity.

The crackpot index is a first-filter for filtering out a large portion of people, including more crackpots than not. Use of it after you have chosen to entertain a person, and giving some sort of credance to them, is also insensitivity and arrogance.

By engineering standards, you show yourself to be an ivory-tower resident, in that you rarely check your logic against reality, and much of the current discussion is about reality (as in the implementation of base 12 in the real world.).